# Dissolving knot surgered 4-manifolds by classical cobordism arguments

## TL;DR

This paper demonstrates that classical cobordism techniques can quickly show that knot surgered 4-manifolds become standard after stabilization, and that many such manifolds are nearly decomposable, simplifying their classification.

## Contribution

It introduces a unified cobordism-based approach to prove stabilization and decomposability results for knot surgered 4-manifolds, extending classical methods to modern problems.

## Key findings

- Knot surgered 4-manifolds become diffeomorphic to original after stabilization.
- Classical cobordism arguments provide quick proofs of stabilization.
- Many almost completely decomposable 4-manifolds are shown to be nearly decomposable.

## Abstract

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold X_K becoming diffeomorphic to X after a single stabilization by connected summing with S^2 x S^2 or CP^2 # -CP^2, and almost complete decomposability of X_K for many almost completely decomposable X, such as the elliptic surfaces.